Curiously enough, scroll and scrawl are not related but have become so through modern semantic marriage, which makes inlaws of many other words related to writing and undignified physical activity.
Scrawl is possibly related to crawl and sprawl; the former is a form of abject movement associated with "lower" life forms or human misery, the latter, with undignified posture, whether the body is at leisure and unguarded or cut down unexpectedly as from a bullet. But note the possible relationship to "scribble" through "scribble-scrabble," and note also the childlike nature of these words in relation to the childlikeness of the names of weaving and textile implements discussed earlier.
If you read to the end of the etymological entry below, you will find a reference to women's genitalia.
1610s, perhaps from M.E. scrawlen "spread out the limbs, sprawl" (late 14c.), which possibly is an alteration of sprawlen (see sprawl) or crawl. Related: Scrawled; scrawling. The noun is recorded from 1690s.
1530s, "to scrawl, scribble," from Du. schrabbelen, frequentative of schrabben "to scratch," from the same root as scrape (q.v.). Meaning "to struggle, scramble" first recorded 1630s. The game Scrabble is from 1950, proprietary name (reg. U.S.), probably from scribble-scrabble "hasty writing" (1580s), a reduplication of scribble.
"scrawl aimlessly," 1935, from dial. doodle, dudle "fritter away time, trifle," or associated with dawdle. It was a noun meaning "simple fellow" from 1620s.
LONGFELLOW: That's a name we made up back home for people who make foolish designs on paper when they're thinking. It's called doodling. Almost everybody's a doodler. Did you ever see a scratch pad in a telephone booth? People draw the most idiotic pictures when they're thinking. Dr. Von Holler, here, could probably think up a long name for it, because he doodles all the time. ["Mr. Deeds Goes to Town," screenplay by Robert Riskin, 1936; based on "Opera Hat," serialized in "American Magazine" beginning May 1935, by Clarence Aldington Kelland]
Related: Doodled; Doodling.
Doodle Sack. A bagpipe. Dutch. -- Also the private parts of a woman. ["Classical Dictionary of the Vulgar Tongue," 1796]